Suppose the utility function of a consumer is U=X^aY^b where X and Y are respectively the quantity of commodities X and Y, and a and b are positive constraints such that a + b=1. When U is maximized, being subject to the budget constraint, pX+qY=Z, where p,q and Z signify respectively the price of X, the price of Y, and the given income of the consumer, express X and Y as functions ofp, q and Z.

Show that the ratio of expenditure on each commodity to the aggregate expenditure is a given constant. Draw the indifference map between X and Y in the case of the utility function given in the problem and show that it is a homothetic case where Engel curve is a straight line through the origin.